I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.

Help anyone?

Printable View

- March 28th 2007, 02:18 AMOReillyFind point M (vectors)
I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.

Help anyone? - March 28th 2007, 04:13 AMThePerfectHacker
- March 28th 2007, 05:08 AMOReilly
- March 30th 2007, 07:53 AMearboth
Hello, OReilly,

I've attached an image which show you what to do in 4 steps. The steps are numbered.

1. Construct the midpoint of AD, call it P.

2. Construct the midpoint of BC, call it Q.

3. Construct the midpoint of PQ: That's M.

If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

In step #4 you see the complete result.

Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

EB - March 30th 2007, 08:59 AMOReilly
- March 31st 2007, 12:58 AMearbothone final remark
Hello, OReilly,

if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

If

A(a1 , a2)

B(b1 , b2)

C(c1 , c2)

D(d1 , d2)

then

M(¼ (a1+b1+c1+d1), ¼ (a2+b2+c2+d2))

EB - March 31st 2007, 03:50 AMOReilly